Pursuit–Evasion Problem of Unmanned Surface Vehicles in a Complex Marine Environment

Abstract

In order to solve the interception problem of escaping targets under environmental interference and inaccurate target perception, this paper introduces the guidance interception algorithm and introduces a Kalman filter to predict the trajectory of escaping targets. In order to make a USV pursuit under a disturbance in a time-varying real marine environment, a disturbance observer is proposed to estimate the external disturbance and model uncertainty. The simulation of pursuit–evasion results shows that the interception algorithm designed enables the pursuer to intercept the escaping target in a time-varying, disturbed marine environment. Both the simulation and experimental results are provided to validate the effectiveness of the method.

1. Introduction

In recent years, with the continuous development of intelligent control theory and with continuous improvements to actual demand, unmanned surface vehicles (USVs) have been considered for a wide range of potential applications in ocean exploration, coastal defense, and intelligent cruises [1,2]. Under an efficient communication network technology [3], there has been a lot of work and focus on the path tracking control of USVs, such as environmental monitoring, hydrological measurement, and path tracking [4,5]. Since the pursuit–evasion problem involves many hot issues in control fields, this issue has always been the focus of intelligent systems [6,7]. Nevertheless, the pursuit–evasion of USVs is still an open problem that needs to be overcome.

In the implementation of target interception, Ref. [8] intercepted an escaping target based on the initial position and relative orientation of the tracker and the escaping target. The authors of [9] achieved the capture of an escaping target, but the motion model assumed in that article only allows the evader and the hunter to move in the horizontal or vertical directions, which is obviously inconsistent with actual motion. In addition, a method based on Reinforcement Learning [10] was gradually applied to solve the pursuit–evasion problem and allowed multiple robots to track a single target. Ref. [11] encircled escaping targets with multiple hunters in an environment of obstacles. By establishing multiple encirclement points around the target, multiple pursuers blocked the escape route of the evader in all directions [12]. Additionally, the encirclement strategy is often used to encircle static targets [13]. However, once the trajectory of the dynamic target becomes complex, the applicability of this algorithm becomes questionable. Although more pursuers can ensure a higher success rate of hunting, the strategy of multiple pursuers is based on tracking and interception by a single pursuer.

An accurate target location is of great significance for intercepting the escaping target. In article [14], a target position estimator was proposed, and a corresponding control law was designed. However, it cannot be ignored that the escaping target has high mobility. Moreover, the interception direction of the pursuer should be time-varying; that is, it should be changed according to the movement of the target. The Kalman filter is widely used because it can predict the motion and position of the target simply and conveniently. To intercept the escaping target more accurately, the Kalman filter algorithm is adopted in this paper to track and predict an escaping target’s trajectory. For target tracking, the algorithm based on line of sight (LOS) is generally used in the marine field [15]. However, the LOS cannot predict the target trajectory in advance for interception. As a simple and effective interception algorithm, the missile guidance law is widely used in the aviation field. Therefore, the missile guidance law is adopted to intercept escaping vessels in the marine field.

When performing tasks in an actual environment, an USV will be affected by environmental interference by the wind, waves, and currents in a marine environment [16] and by threats from moving obstacles such as sailing ships [17]. A small USV will be more sensitive to external disturbances due to its small main scale and displacement [18]. Although Ref. [19] carried out sea trials of an USV and avoided obstacles and rescued people from drowning, it did not consider interference from the external environment. How to deal with uncertain nonlinearity and interference is an important problem in target tracking and interception [20]. However, the above article does not consider the uncertainty of the environment, which brings challenges to the interception of dynamic escaping targets.

Previous studies on the pursuit–evasion problem had several shortcomings. First, many studies set the escaping target as static, which is not consistent with pursuit–evasion movement in an actual scenario. Second, in order to validate the effectiveness of the interception strategy, only the path of the interceptor is optimized, and the motion of the escaping target is simplified to linear motion, which is also inconsistent with an actual scene. Finally, at present, there are few experiments on the pursuit–evasion problem and the interception strategy is only validated by simulation.

In view of the above challenges, we aim to design a strategy that can intercept escaping targets. Compared with previous work, the contribution of this paper is mainly reflected in the following three aspects. Firstly, considering the dynamic curve motion of the target, a surface interception strategy is constructed to intercept fast maneuvering targets. Secondly, this paper considers predictability based on the Kalman filter, which can improve the efficiency of interception. Thirdly, a disturbance observer is proposed, for which the accuracy of estimating a disturbance is independent from the designed pursuit algorithm.

The rest of this article is organized as follows: in the Section 2, the basic assumptions of the pursuit–evasion problem are introduced, and the guidance algorithm is adopted to solve the interception problem of evaders. The kinematics and dynamics of USV and the Kalman filter algorithm are also given. In the Section 3, the controller and disturbance observer are designed, and a stability analysis of the system is presented. A simulation of the pursuit algorithm is presented in the Section 4. The Section 5 provides the experimental results. Finally, some conclusions are drawn in the Section 6.

2. Problem Formulation and Tracking Strategy

2.1. Pursuit–Evasion Problem Statement

Suppose that there are pursuit USVs and escaping target USVs in a sea area. When the escaping target ship enters the detection range of the pursuit USV, the interceptor will exhibit an interception behavior. The target also has the ability to perceive the pursuer; when the target USV senses the surrounding captors, it will change its route. The pursuer cannot determine the escape strategy of the target and can only adjust its interception direction in real time according to the perceived position and speed of the target.

2.2. Guidance Method in Pursuit–Evasion Games

Suppose at time $k$ that the pursuer $P(x_{}^k,y_{}^k)$ has perceived the escaping target $E(x_E^k,y_E^k)$, and the line $l$ formed by the pursuer pointing to the escaping target $E$ is called the line of sight. The angle formed by the line of sight and the x-axis is the line of sight angle. The angle between the target speed $V_E$ and $l$ is ${\psi }_E$, and the angle between the pursuer speed and $l$ is ${\psi }_P$. The angle between the line connecting $P$ and the virtual hunting point $E^{\prime }$, and the x-axis is the desired heading angle ${\psi }_d$. A schematic diagram of the guidance method is shown in Figure 1.

The relative motion equation of the pursuer and the evader is as follows:

$$\{\begin{array}{l}\dot{l}\hspace{0.17em}=V_P\cos {\psi }_p-V_E\cos {\psi }_E\\ l\dot{\theta }=V_P\sin {\psi }_p\hspace{0.17em}-V_E\sin {\psi }_E\end{array}$$

(1)

The normal speed of the pursuer needs to be equal to the escaping target, that is $\dot{\theta }=0$; substituting it into the Formula (1), we have the following:

$$V_P\sin {\psi }_p=V_E\sin {\psi }_E$$

(2)

According to the geometric relationship, the desired heading angle is as follows:

$${\psi }_d={\sin }^{-1}(\frac{V_E}{V_P}\sin {\psi }_E)+\theta$$

(3)

2.3. Kinematics and Dynamics of USV

A reasonable mathematical model of motion is the basis of ship motion simulation and control algorithm design. The motion model of USV is simplified to a three degree of freedom (DOF) motion model of the water surface. The motion needed to be taken into consideration is in the horizontal plane, including the surge, the sway, and the yaw. The vessel motion model in the horizontal plane is shown in Figure 2.

The model of a USV with three DOFs can be described as follows:

$$\{\begin{array}{l}\dot{x}=u\cos \psi -v\sin \psi \\ \dot{y}=u\sin \psi +v\cos \psi \\ \dot{\psi }=r\end{array}$$

(4)

In the above expressions, $x$, $y$, and $\psi$ denote the position and orientation in the earth-fixed coordination system, and $u$, $v$, and $r$ are the velocity vectors of the USV with respect to the body-fixed frame, which represent the surge, the sway, and the yaw.

The dynamic equation of the USV when encountering an external disturbance is as follows:

$$\{\begin{array}{l}\dot{u}=\frac{m_{22}}{m_{11}}vr-\frac{d_{11}}{m_{11}}u+\frac{1}{m_{11}}\left({\tau }_u+{\tau }_{wu}\right)\\ \dot{v}=-\frac{m_{11}}{m_{22}}ur-\frac{d_{22}}{m_{22}}v+\frac{1}{m_{22}}\left({\tau }_v+{\tau }_{wv}\right)\\ \dot{r}=\frac{m_{11}-m_{22}}{m_{33}}uv-\frac{d_{33}}{m_{33}}r+\frac{1}{m_{33}}\left({\tau }_r+{\tau }_{wr}\right)\end{array}$$

(5)

In the above expressions, $m_{ii}$ is the positive definite inertia matrix and $d_{ii}$ is the damping matrix. ${\tau }_{ui}$ and ${\tau }_{ri}$ represent the longitudinal force and yaw moment of the USV, and ${\tau }_{wu}$, ${\tau }_{wv}$, and ${\tau }_{wr}$ represent the influence of an external time-varying uncertain disturbance on the USV motion.

2.4. Kalman Filter Algorithm

In this paper, the Kalman filter algorithm is used to estimate and predict the position of the target in real time, and the target state is constantly updated during a pursuit. The dynamic equation and measurement equation of the discrete-time system of the Kalman filter can be expressed as follows:

$$\{\begin{array}{l}{\widehat{\mathit{x}}}_k^{-}=\mathit{F}{\widehat{\mathit{x}}}_{k-1}^{-}+\mu \mathit{\eta }\\ {\mathit{Z}}_k^{}=\mathit{H}{\mathit{X}}_k^{}+\mathit{\omega }\end{array}$$

(6)

where ${\widehat{\mathit{x}}}_k^{-}$ refers to the state estimation at time $k-1$ and $\mathit{F}$ represents the state transition matrix. $\mathit{\eta }$ is a white noise series subject to Gaussian distribution with a mean value of zero and a positive definite covariance matrix of $\mathit{Q}$. $\mu$ is the process noise gain coefficient. ${\mathit{Z}}_k$ is the observed measurement of the system, $\mathit{H}$ is the observation coefficient matrix, and $\omega$ is the white noise series subject to Gaussian distribution with zero mean and positive definite covariance matrix. $\mathit{R}$, $\mathit{\eta }$, and $\mathit{\omega }$ are independent from each other.

The equation for the covariance matrix of the state vector at the predicted state time is as follows:

$${\mathit{P}}_k^{-}=\mathit{F}{\mathit{P}}_{k-1}^{-}{\mathit{F}}^T+\mathit{Q}$$

(7)

where ${\mathit{P}}_k^{}$ represents the error covariance matrix at time $k$. The updated prediction state and noise covariance matrix with observations is as follows:

$${\mathit{K}}_k^{}={\mathit{P}}_k^{-}{\mathit{H}}^T{(\mathit{H}{\mathit{P}}_k^{-}{\mathit{H}}^T+\mathit{R})}^{-1}$$

(8)

where ${\mathit{K}}_k$ represents the gain of the filter at time $k$, which is the intermediate calculation result of the filter. $\mathit{R}$ is the measurement noise, and its value can be directly set according to the accuracy of the observation equipment.

The updated prediction state and noise covariance matrix with observations is:

$$\{\begin{array}{l}{\widehat{\mathit{x}}}_k^{}={\widehat{\mathit{x}}}_k^{-}+{\mathit{K}}_k^{}({\mathit{Z}}_k^{}-\mathit{H}{\widehat{x}}_k^{-})\\ {\mathit{P}}_k^{}=({\mathit{I}}_{4\times 4}^{}-{\mathit{K}}_k^{}\mathit{H}){\mathit{P}}_k^{-}\end{array}$$

(9)

where $({\mathit{Z}}_k^{}-\mathit{H}{\widehat{x}}_k^{-})$ represents the residual error of the actual observation and the predicted observation, which will correct the a priori value together with the Kalman gain. ${\mathit{P}}_k^{-}$ represents the covariance of a priori estimation of ${\widehat{\mathit{x}}}_k^{-}$ at time k, and ${\mathit{P}}_k^{}$ represents the covariance of the posterior estimate of ${\widehat{\mathit{x}}}_k^{}$. $\mathit{P}$ represents the error covariance, and a smaller value represents a more accurate prediction of the state. The initial value only affects the initial convergence speed, and the value of $\mathit{P}$ will be updated with the iteration of the Kalman filter. Set initial value ${\mathit{P}}_0=0$.

The motion equation of the escaping target object can be expressed as follows:

$$\{\begin{array}{l}{x}_k^{}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=x_{k-1}^{}+v_{k-1}^{x}T\\ v_{k-1}^x=v_{k-1}^x+{\alpha }_{k-1}^{x}T\\ y_k^{}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=y_{k-1}^{}+v_{k-1}^{y}T\\ v_{k-1}^y=v_{k-1}^y+{\alpha }_{k-1}^{y}T\end{array}$$

(10)

where $x_{k-1}^x$, $v_{k-1}^x$, and ${\alpha }_{k-1}^x$ are the position, velocity, and acceleration of the target in the x-axis direction at time $t_{k-1}$, and $y_{k-1}^{}$, $v_{k-1}^y$, and ${\alpha }_{k-1}^y$ are the position, velocity, and acceleration of the target in the y-axis direction at time $t_{k-1}$. $T$ is the time interval between two adjacent sampling points. When the sampling frequency is approximately 10 Hz, the value of $T$ can be taken as 0.1 s. The above formula in matrix form can be written as follows:

$$\left[\begin{array}{l}{x}_k^{}\\ v_k^x\\ y_k^{}\\ v_k^y\end{array}\right]\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\left[\begin{array}{cccc}1& T& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& T\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{l}{x}_{k-1}^{}\\ v_{k-1}^x\\ y_{k-1}^{}\\ v_{k-1}^y\end{array}\right]\hspace{0.17em}+\left[\begin{array}{cc}0.5T^2& 0\\ T& 0\\ 0& 0.5T^2\\ 0& T\end{array}\right]\left[\begin{array}{l}{\alpha }_{k-1}^x\\ {\alpha }_{k-1}^x\\ {\alpha }_{k-1}^y\\ {\alpha }_{k-1}^y\end{array}\right]\hspace{0.17em}$$

(11)

The dynamic noise vector of the system is as follows:

$$\mathit{\eta }={\left[0.5{\alpha }_{k-1}^x{T}^2,{\alpha }_{k-1}^{x}T,0.5{\alpha }_{k-1}^y{T}^2,{\alpha }_{k-1}^{y}T\right]}^T$$

(12)

After the above system state equation and observation equation are established, the Kalman filter equation can be used to continuously predict the position of the target at the next time.

$$\mathit{H}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\end{array}\right]$$

(13)

3. Controller Design of USV

3.1. Disturbance Observer Design

The external environmental disturbance is treated as the combined effect of the uncertainty of the model and disturbances by the wind, waves, and the current in the external marine environment. The environmental disturbance force can be expressed as follows:

$${\tau }_E={[{\tau }_w{}_u {\tau }_{wv} {\tau }_{wr}]}^T$$

(14)

The disturbance is regarded as an uncertain time-varying disturbance with upper bound and slow variation. ${\tau }_{wu}$, ${\tau }_{wv}$, and ${\tau }_{wr}$ represent the influence of ocean disturbances and model uncertainty acting on the surge, sway, and yaw directions, respectively.

In this paper, the disturbance in the surge and yaw directions was estimated online, and the trajectory tracking controller with disturbance compensation was designed in combination with the output of the disturbance observer:

$$\{\begin{array}{l}{\hat{\mathit{\tau }}}_{\mathit{e}}=\mathit{\mu }+{\mathit{K}}_e\mathit{M}\mathit{\upsilon }\\ \dot{\mathit{\mu }}={\mathit{K}}_e\left(\mathit{C}\left(\mathit{\upsilon }\right)\mathit{\upsilon }+\mathit{D}\left(\mathit{\upsilon }\right)\mathit{\upsilon }-\mathit{\tau }-{\mathit{K}}_{e}M\mathit{\nu }\right)-{\mathit{K}}_e\mathit{\mu }\end{array}$$

(15)

where ${\widehat{\mathit{\tau }}}_{\mathrm{e}}={\left[{\widehat{\tau }}_{eu},{\widehat{\tau }}_{ev},{\widehat{\tau }}_{er}\right]}^{{\rm T}}$ is introduced to represent the external estimation and ${\mathit{K}}_{\mathrm{e}}={\left[{\mathit{K}}_{eu},{\mathit{K}}_{ev},{\mathit{K}}_{er}\right]}^{{\rm T}}$ is the disturbance gain.

The estimation error of each degree of freedom disturbance variable is defined as follows:

$$\{\begin{array}{l}{\tilde{\tau }}_{wu}={\tau }_{wu}-{\widehat{\tau }}_{wu}\\ {\tilde{\tau }}_v\hspace{0.17em}\hspace{0.17em}={\tau }_v\hspace{0.17em}\hspace{0.17em}-{\widehat{\tau }}_v\hspace{0.17em}\\ {\tilde{\tau }}_{wr}={\tau }_{wr}-{\widehat{\tau }}_{wr}\end{array}$$

(16)

The Lyapunov function is designed as follows:

$$V_{wu}^{}=\frac{1}{2}{\tilde{\tau }}_{wu}^2$$

(17)

Taking the derivation of both sides of Equation (17) and substituting Equation (15) into it, we obtain the following:

$$\begin{array}{l}{\dot{V}}_{wu}^{}={\tilde{\tau }}_{wu}^{}{\dot{\tilde{\tau }}}_{wu}^{}={\tilde{\tau }}_{wu}^{}({\dot{\tau }}_{wu}^{}-{\dot{\widehat{\tau }}}_{wu}^{})\\ ={\tilde{\tau }}_{wu}^{}{\dot{\tau }}_{wu}^{}+k_{wu}^{}{\tilde{\tau }}_{wu}^{}\{\mathit{C}(\mathit{\upsilon })\mathit{\upsilon }+\mathit{D}(\mathit{\upsilon })\mathit{\upsilon }-\mathit{\tau }-k_{wu}^{}\mathit{M}\mathit{\upsilon }+\mathit{\mu }+\mathit{M}\dot{\mathit{\upsilon }}\}\\ =-k_{wu}^{}{\tilde{\tau }}_{wu}^2\\ \le 0\end{array}$$

(18)

Therefore, the longitudinal disturbance observation error ${\widehat{\tau }}_{wu}$ is globally exponentially stable, and similarly, the disturbance observations ${\widehat{\tau }}_{wv}$ and ${\widehat{\tau }}_{wr}$ can also accurately estimate the actual disturbance. The disturbance observer and control algorithm designed are independent from each other; the accuracy of disturbance estimation does not affect the controller design.

3.2. Heading Control

In the control strategy, define the angle tracking error as follows:

$$\tilde{\psi }=\tilde{\psi }-{\psi }_d$$

(19)

In order to adopt the first-order sliding mode to converge the heading tracking angle error, it is necessary to obtain the yaw rate error of the USV. The yaw rate error $\tilde{r}$ can be obtained by the time derivative of Equation (19):

$$\tilde{r}=r-r_d$$

(20)

Considering the fast convergence of sliding mode control, the first-order sliding surface is described by the following formula:

$$S_{\phi r}=\dot{\tilde{\psi }}+{\lambda }_{\phi }\tilde{\psi }=r-r_d+{\lambda }_{\phi }\tilde{\phi }$$

(21)

where ${\lambda }_{\phi r}$ is a positive constant and it represents the sliding mode design variable.

The Lyapunov function considering heading angle and external disturbance can be expressed as follows:

$$V_{\phi r}=\frac{1}{2}{S}_{\phi }^2+\frac{1}{2}{\tilde{\tau }}_{wr}^2$$

(22)

By differentiating the above formula, we obtain the following:

$$\begin{array}{l}{\dot{V}}_{\phi r}=S_{\phi }{\dot{S}}_{\phi }+{\tilde{\tau }}_r^2{\dot{\tilde{\tau }}}_r^{}\\ \hspace{0.17em}=S_{\phi }\left(\dot{r}-{\dot{r}}_d+{\lambda }_{\phi }\left(r-r_d\right)\right)+{\tilde{\tau }}_r^2{\dot{\tilde{\tau }}}_r^{}\\ \hspace{0.17em}=S_{\phi }\left(\frac{m_{11}-m_{22}}{m_{33}}uv-\frac{d_{33}}{m_{33}}r+\frac{1}{m_{33}}({\tau }_r+{\tau }_{wr})-{\dot{r}}_d\right)\\ \hspace{1em}\hspace{0.17em}\hspace{1em}+S_{\phi }\left({\lambda }_{\phi }\left(r-r_d\right)\right)+{\tilde{\tau }}_r^2{\dot{\tilde{\tau }}}_r^{}\end{array}$$

(23)

To ensure that the pursuer can hunt the target more effectively, the heading angle controller is designed as follows:

$${\tau }_r=-m_{33}(\frac{m_{11}-m_{22}}{m_{33}}uv-\frac{d_{33}}{m_{33}}r+{\widehat{\tau }}_{er})+{\dot{r}}_d-{\lambda }_{\phi }(r-r_d)-k_{\tau r}{S}_{\phi r}$$

(24)

where $k_{\tau r}$ is a positive constant and it represents the heading angle gain parameter.

Substituting the above formula into Formula (24), we obtain the following:

$${\dot{V}}_{\phi r}=-k_{\tau r}{S}_{\phi }^2-k_{wr}^{}{\tilde{\tau }}_r^2{\dot{\tilde{\tau }}}_r^{}\le 0$$

(25)

It can be known that the sliding surface $S_{\phi r}$ will converge to zero in finite time and that the heading angle error will also converge to zero, which means $\underset{t\to \infty }{\lim }\left(\tilde{\psi },\tilde{r}\right)=\left(0,0\right)$.

3.3. Surge Control

Longitudinal velocity tracking error variables can be defined as follows:

$$\underset{t\to \infty }{\lim }u\to u_d$$

(26)

In the pursuit problem, $u_d$ is the pursuer’s maximum speed. The surge following error $u_e$ is described as follows:

$$u_e=u-u_d^{}$$

(27)

The Lyapunov function considering longitudinal thrust and external disturbance is designed as follows:

$$V_{ue}=\frac{1}{2}{u}_e{}^2+\frac{1}{2}{\tilde{\tau }}_{wu}^2$$

(28)

Taking the derivative of the above formula, we obtain the following:

$${\dot{V}}_{ue}=u_e(\frac{m_{22}}{m_{11}}vr-\frac{d_{11}}{m_{11}}u+\frac{1}{m_{11}}\left({\tau }_u+{\tau }_{eu}\right)-{\dot{u}}_d)+{\tilde{\tau }}_{wu}^{}{\dot{\tilde{\tau }}}_{wu}^{}$$

(29)

In order to ensure the convergence of the velocity term, the design surge control law for ${\tau }_u$ is chosen as follows:

$${\tau }_u=m_{11}(-k_u{u}_e-\frac{m_{22}}{m_{11}}v{r}_{}+\frac{d_{11}}{m_{11}}u+{\dot{u}}_d^{})-{\widehat{\tau }}_{eu}$$

(30)

where $k_u$ is a positive constant. If the surge force designed in Formula (31) can ensure the function value is always negative, then the actual velocity of the USV will be consistent with the desired value. Substituting the above expression into Formula (29), we have the following:

$${\dot{V}}_{ue}=-k_u{u}_e{}^2-k_{wu}^{}{\tilde{\tau }}_{wu}^2<0$$

(31)

According to Formula (31), the velocity error will gradually converge to zero, and the velocity of the pursuer will reach the desired rate. When the direction and velocity of the pursuit USV reach the desired value and converge continuously, the escaping target can be intercepted by the pursuer.

4. Simulation and Analysis

In order to validate the effectiveness of intercepting escaping targets under the guidance strategy proposed in this paper, a simulation experiment was carried out for the algorithm. The ship model used in the simulation is the scale model of CyberShip II (CS2) [21], which is the test tool of the Marine Cybernetics Laboratory (MCLab).

The hydrodynamic parameters are also taken from the parameters of the well-known surface vehicle CS2. In order to test the robustness of the proposed control algorithm to external disturbances, it is assumed that there is an error of 10% between the nominal value and the actual value of the motion parameters. $\stackrel{⌢}{•}$ indicates the nominal value and the specific nominal value of hydrodynamic parameters, which can be found in the literature [21]. The actual values of the motion parameters are shown in

Table 1. Actual values of CS2.

Parameters	Value	Parameters	Value
$m_{11}$	$1.1{\widehat{m}}_{11}$	$d_{11}$	$1.1{\widehat{d}}_{11}+0.2{\widehat{d}}_{11}\left|u\right|$
$m_{22}$	$1.1{\widehat{m}}_{22}$	$d_{22}$	$1.1{\widehat{d}}_{22}+0.2{\widehat{d}}_{22}\left|v\right|$
$m_{23}$	$1.1{\widehat{m}}_{23}$	$d_{23}$	$1.1{\widehat{d}}_{23}$
$m_{33}$	$1.1{\widehat{m}}_{33}$	$d_{33}$	$1.1{\widehat{d}}_{33}+0.2{\widehat{d}}_{33}\left|r\right|$

.

The parameter model of USV was taken as a constant disturbance and summed up in the disturbance model for external wind, waves, and currents. The environmental disturbance model is defined as follows:

$$\{\begin{array}{l}{\tau }_{wu}=(h(s)w_u(s)+{\xi }_{wu})m_{11}\\ {\tau }_{wv}=(h(s)w_v(s)+{\xi }_{wv})m_{22}\\ {\tau }_{wr}=(h(s)w_r(s)+{\xi }_{wr})m_{33}\end{array}$$

(32)

with

$$h(s)=\frac{K_{\omega }s}{s^2+2\zeta {\omega }_{0}s+{\omega }_0^2}$$

(33)

where $w_u$, $w_v$, and $w_r$ are Gaussian white noise; ${\xi }_{wu}$, ${\xi }_{wv}$, and ${\xi }_{wr}$ are the uncertainty error coefficient of the model. $\zeta =0.3$, $K_{\omega }=0.25$, ${\sigma }_m=0.4$, ${\omega }_0=0.8$, and $k_{wu}=k_{wv}=k_{wr}=0.1$. The maximum values of ${\tau }_{eu}/m_{11}$, ${\tau }_{ev}/m_{22}$, and ${\tau }_{er}/m_{33}$ are 0.1 m/s², 0.1 m/s², and 0.5 m/s², respectively.

The recommended values of parameters in the controller in the simulation are shown in

Table 2. USV control parameters.

Parameters	Value	Parameters	Value
$k_{eu}$	10	${\lambda }_{\phi }$	0.5
$k_{ev}$	10	$k_{\tau r}$	2
$k_{er}$	10	${\xi }_{wu}$	0.1
$k_{wr}$	2	${\xi }_{wv}$	0.1
$k_{wu}$	2	${\xi }_{wr}$	0.1
$k_u$	10

.

The initial state of the pursuer is cruising. When the escaping target enters its sensing range, the pursuer adjusts its state and begins its pursuit. The escaping target moves along a curved path. The curved path can be parameterized as follows:

$$\{\begin{array}{l}(x_E,y_E)=(0.4\omega +10 , 0.6\omega +10) ,0\le \omega <100\\ (x_E,y_E)=(0.6\omega -10, 20\sin (0.2(\omega /25-4))+70) , \omega \ge 100\end{array}$$

(34)

where the path parameter is $\omega =V_E/\sqrt{x_E^{\prime }{}^2+y_E^{\prime }{}^2}$. Within the range of simulation time t = 0–170 s, the trajectories of the pursuer and evader are shown in Figure 3.

Set the parameters in the filter as the parameters in the actual observer to maintain the consistent performance for the actual observer. $\mathit{H}$ is the measurement vector of the sensor, which is the coordinate value obtained through the UWB in this system. As the source of the systematic observation data, the UWB positioning equipment has a coordinate accuracy of 0.1 m, so the element value in matrix $\mathit{R}$ was taken as 0.1. The element value in matrix $\mathit{Q}$ was taken as 0.1.

The observation trajectory obtained by the observation station is obviously oscillating, but the observation noise value of the observer is relatively small, so it is closer to the value of the real trajectory. This measurement value can be directly used for target-locking and interception in the simulation. However, since the actual observer may have abnormal values, it is necessary to determine and discard abnormal values. The position errors before and after filtering are shown in Figure 4.

The detection range of the pursuer is a circular area with a radius of 20 m. When the escaping target enters the pursuer’s perceptual range around t = 23 s, the pursuer adjusts the direction of movement and performs the pursuit. However, due to the limitation of the minimum turning radius of the pursuit USV and the influence of the evader’s rapid adjustment of the direction of motion, there will be a short deviation in the route when the pursuer tracks the evader in real time.

Figure 5 shows the angle changes in the pursuer and evader. When the evader enters the perceptual range of the pursuer around t = 23 s, the pursuer adjusts the motion angle to intercept the target.

From Figure 6, it can be seen that in the process of performing the pursuit, although the evader adopts a variety of motion modes, such as linear motion and curve motion, the pursuer always maintains a good tracking effect. It can be seen that in about 150 s, the distance between the pursuer and the escaping target is less than 1 m. This is enough to ensure successful interception of the escaping target by the pursuit USV.

In order to test the estimation effect of the disturbance observer on the uncertainty of the system, the actual value of the total disturbance momentum of the system in this simulation is compared with the estimated value. Because the USV belongs to the underactuated control system, only the designed thrust disturbance and rudder force disturbance are compared. The observation results of system disturbance are shown in Figure 7 and Figure 8.

From the disturbance observation results, it can be observed that the estimated values of system disturbances ${\tau }_{wu}$ and ${\tau }_{wr}$ fit the expected values more accurately. The designed disturbance observation algorithm has a good estimation effect on unknown system parameter perturbation and the time-varying external disturbance force.

5. Experimental Results

In order to further validate the effectiveness of the interception algorithm proposed in this paper, the pursuit-evasion experiment ispresented in this section. Two USVs with the same principal dimensions are defined as the evader (yellow) and the pursuer (orange). The mass of the USV is 12 kg, the length is 0.8 m, and the width is 0.24 m. The maximum speed of the evader and the pursuer are 0.5 m/s and 0.6 m/s, respectively. The position is measured by the ultra-wideband (UWB), and the positioning accuracy is 0.1 m. In addition, the pursuer is equipped with cameras to hunt the evader. The composition of the USV is shown in Figure 9.

The UWB position measurement system is composed of three base station modules at fixed positions, and the final positioning data are generated only on the “base station” module connected to the controller. The “label” module is fixed on the moving ship. The “tag” can be easily added or reduced. Figure 10 shows the structure of UWB position.

UWB is used to obtain and predict the position of the pursuit vehicle and the target vehicle through the Kalman filtering algorithm. Since the movement speed of the pursuer and target in this test changes slowly, and the sampling frequency of UWB is greater than 10 Hz, the ship can be regarded as moving at a uniform speed within the sampling frequency. Due to the noise and error of the data directly obtained by the UWB sensor in real time, the data need to be processed by the filtering algorithm. First, obviously wrong UWB data are eliminated. Data points beyond the scope of the tested water area or beyond the radius of 2 m of the current position obtained by the least squares method will be regarded as abnormal measurement data points. Then, the least squares method is used to perform second-order fitting on 20 data points that were recently saved. The fitting result is taken as the prediction value, and the Kalman filter algorithm is used to correct the latest measurement value to obtain the optimal target motion information.

In the test, the evader always moves at a heading angle of 10°. The pursuer senses the target and executes an interception in order to show the trajectories of the evader and the pursuer more clearly. Figure 11 shows their trajectory curves according to the UWB data. Although the escaping target moves along a curve, it can still be seen that the proposed encirclement algorithm can intercept the escaping target.

6. Conclusions

Combining the guidance algorithm and the underactuated characteristics of a USV, this paper proposes an interception algorithm for escaping targets. For the escaping target with a flexible escape route, the algorithm can still achieve rapid interception. According to the Lyapunov stability theory, it is validated that the designed controller can ensure the convergence of the motion error of USV in a complex marine environment. Finally, the simulation and experimental results validate the effectiveness of the interception algorithm. In this paper, the uncertainty of the environment and the flexibility of the escaping target are comprehensively considered. In addition, the missile guidance method is applied to the interception field, and the trajectory of the escaping target is further predicted by combining the Kalman algorithm.

However, this study still has several shortcomings. First, although the controller considering uncertain external disturbance is designed in the algorithm, due to the lack of equipment, this function is not realized in the experiment. Second, the route of the escaping target should be adjusted in real time according to the state of the pursuer. In this paper, the escaping target moves according to the predefined route in advance. In addition, obstacles are an important factor affecting interception. In future research, we plan to consider the above shortcomings. Future work will further consider the game strategy between the evader and the pursuer, and the obstacle avoidance algorithm will be applied to the pursuit algorithm in more complex environments.